Impossibility of General, Dominant-Strategy Implementation Game Theory Course: Jackson, Leyton-Brown & Shoham Game Theory Course: Jackson, Leyton-Brown & Shoham Impossibility of General, Dominant-Strategy Implementation .
So, if we are considering implementation in dominant strategies, it is enough to look only at social choice functions for which truth is a dominant strategy: the set of non-manipulable or strategy-proof social choice functions. . Dominant Strategies and Mechanisms: Let us apply the revelation principle Consider a society N, O and any mechanism A, M for which every agent has a dominant strategy for each preference. There exists a social choice function C (a “direct mechanism”) for which truthful announcement of preferences is a dominant strategy. Game Theory Course: Jackson, Leyton-Brown & Shoham Impossibility of General, Dominant-Strategy Implementation .
. Dominant Strategies and Mechanisms: Let us apply the revelation principle Consider a society N, O and any mechanism A, M for which every agent has a dominant strategy for each preference. There exists a social choice function C (a “direct mechanism”) for which truthful announcement of preferences is a dominant strategy. So, if we are considering implementation in dominant strategies, it is enough to look only at social choice functions for which truth is a dominant strategy: the set of non-manipulable or strategy-proof social choice functions. Game Theory Course: Jackson, Leyton-Brown & Shoham Impossibility of General, Dominant-Strategy Implementation .
So, any non-dictatorial social choice function on a full domain of preferences and with at least three alternatives will be manipulable by some agents for some preference profiles. . Impossibility Result . Theorem (Gibbard–Satterthwaite) . Consider a social choice function C : L n �→ O . Suppose that 1. there are at least three outcomes so that | O | ≥ 3 , and 2. C is onto ; that is, for every o ∈ O there is a preference profile [ ≻ ] ∈ L n such that C ([ ≻ ]) = o Truthful reporting of preferences is a dominant strategy for each agent i and each preference ≻ i ∈ L if and only if C is dictatorial: there exists i for whom C ([ ≻ ]) = argmax O ≻ i for all [ ≻ ] ∈ L n . . Game Theory Course: Jackson, Leyton-Brown & Shoham Impossibility of General, Dominant-Strategy Implementation .
. Impossibility Result . Theorem (Gibbard–Satterthwaite) . Consider a social choice function C : L n �→ O . Suppose that 1. there are at least three outcomes so that | O | ≥ 3 , and 2. C is onto ; that is, for every o ∈ O there is a preference profile [ ≻ ] ∈ L n such that C ([ ≻ ]) = o Truthful reporting of preferences is a dominant strategy for each agent i and each preference ≻ i ∈ L if and only if C is dictatorial: there exists i for whom C ([ ≻ ]) = argmax O ≻ i for all [ ≻ ] ∈ L n . . So, any non-dictatorial social choice function on a full domain of preferences and with at least three alternatives will be manipulable by some agents for some preference profiles. Game Theory Course: Jackson, Leyton-Brown & Shoham Impossibility of General, Dominant-Strategy Implementation .
use a weaker form of implementation: the result only holds for dominant strategy implementation, not e.g., Bayes–Nash implementation relax the assumption that agents are allowed to have arbitrary preferences and look at more structured settings. . What does this mean? • Having dominant strategies for all agents and possible preferences is infeasible unless we have a dictatorial social choice function. • However, in practice we can circumvent the Gibbard–Satterthwaite theorem in various ways: Game Theory Course: Jackson, Leyton-Brown & Shoham Impossibility of General, Dominant-Strategy Implementation .
relax the assumption that agents are allowed to have arbitrary preferences and look at more structured settings. . What does this mean? • Having dominant strategies for all agents and possible preferences is infeasible unless we have a dictatorial social choice function. • However, in practice we can circumvent the Gibbard–Satterthwaite theorem in various ways: • use a weaker form of implementation: • the result only holds for dominant strategy implementation, not e.g., Bayes–Nash implementation Game Theory Course: Jackson, Leyton-Brown & Shoham Impossibility of General, Dominant-Strategy Implementation .
. What does this mean? • Having dominant strategies for all agents and possible preferences is infeasible unless we have a dictatorial social choice function. • However, in practice we can circumvent the Gibbard–Satterthwaite theorem in various ways: • use a weaker form of implementation: • the result only holds for dominant strategy implementation, not e.g., Bayes–Nash implementation • relax the assumption that agents are allowed to have arbitrary preferences and look at more structured settings. Game Theory Course: Jackson, Leyton-Brown & Shoham Impossibility of General, Dominant-Strategy Implementation .
Single-Peaked domains: median voting or take the max of peaks, or the min of peaks... Trade: Have a private value for buying (or selling) an indivisible good A price is fixed in advance, declare whether willing to buy (sell) at that price ... we will see more shortly. . Settings with Strategy-Proof Social Choice Functions: Game Theory Course: Jackson, Leyton-Brown & Shoham Impossibility of General, Dominant-Strategy Implementation .
Trade: Have a private value for buying (or selling) an indivisible good A price is fixed in advance, declare whether willing to buy (sell) at that price ... we will see more shortly. . Settings with Strategy-Proof Social Choice Functions: • Single-Peaked domains: • median voting • or take the max of peaks, or the min of peaks... Game Theory Course: Jackson, Leyton-Brown & Shoham Impossibility of General, Dominant-Strategy Implementation .
... we will see more shortly. . Settings with Strategy-Proof Social Choice Functions: • Single-Peaked domains: • median voting • or take the max of peaks, or the min of peaks... • Trade: • Have a private value for buying (or selling) an indivisible good • A price is fixed in advance, • declare whether willing to buy (sell) at that price Game Theory Course: Jackson, Leyton-Brown & Shoham Impossibility of General, Dominant-Strategy Implementation .
. Settings with Strategy-Proof Social Choice Functions: • Single-Peaked domains: • median voting • or take the max of peaks, or the min of peaks... • Trade: • Have a private value for buying (or selling) an indivisible good • A price is fixed in advance, • declare whether willing to buy (sell) at that price • ... we will see more shortly. Game Theory Course: Jackson, Leyton-Brown & Shoham Impossibility of General, Dominant-Strategy Implementation .
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